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pset6_soln

# pset6_soln - 6.042/18.062J Mathematics for Computer Science...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science March 15, 2005 Srini Devadas and Eric Lehman Problem Set 6 Solutions Due: Monday, March 28 at 9 PM Problem 1. Sammy the Shark is a financial service provider who offers loans on the fol- lowing terms. • Sammy loans a client m dollars in the morning. This puts the client m dollars in debt to Sammy. • Each evening, Sammy first charges a “service fee”, which increases the client’s debt by f dollars, and then Sammy charges interest, which multiplies the debt by a factor of p . For example, if Sammy’s interest rate were a modest 5% per day, then p would be 1 . 05 . (a) What is the client’s debt at the end of the first day? Solution. At the end of the first day, the client owes Sammy ( m + f ) p = mp + fp dollars. (b) What is the client’s debt at the end of the second day? Solution. (( m + f ) p + f ) p = mp 2 + fp 2 + fp (c) Write a formula for the client’s debt after d days and find an equivalent closed form. Solution. The client’s debt after three days is ((( m + f ) p + f ) p + f ) p = mp 3 + fp 3 + fp 2 + fp. Generalizing from this pattern, the client owes d d mp + fp k k =1 dollars after d days. Applying the formula for a geometric sum gives: d +1 p − 1 − 1 d mp + f · p − 1 Problem 2. Find closed-form expressions equal to the following sums. Show your work. 2 Problem Set 6 (a) n 9 i − 7 i 11 i i =0 Solution. Split the expression into two geometric series and then apply the formula for the sum of a geometric series. n n i n 9 i − 7 i 9 7 i = 11 i 11 − 11 i =0 i =0 i =0 9 n +1 7 n +1 1 − 11 1 − 11 = 9 1 − 11 − n +1 7 1 − 11 n +1 11 9 11 7 11 = + + − 2 · 11 4 · 11 4 (b) n 3 4 i +5 i =1 Solution. Taking the logarithm reduces this product to an easy sum. n 3 log 3 ( Q n 3 4 i +5 ) 3 4 i +5 = i =1 i =1 P n = 3 4 i +5 i =1 = 3 2 n ( n +1)+5 n (c) n ∞ 1 i j 5 / 3 · 1 − 2 j 1 / 3 j =1 i =0 Solution. This fearsome-looking sum is a paper tiger; we just apply the formula for the sum of a geometric series followed by the formula for the sum of an arithmetic series. n 1 i n ∞ 1 j 5 / 3 · 1 − 2 j 1 / 3 = j 5 / 3 · 1 j =1 i =0 j =1 1 − 1 − 2 j 1 / 3 n = 2 j 2 j =1 1 2 n ( n + 2 )( n + 1) = 3 3 Problem Set 6 Problem 3. There is a bug on the edge of a 1-meter rug. The bug wants to cross to the other side of the rug. It crawls at 1 cm per second. However,...
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pset6_soln - 6.042/18.062J Mathematics for Computer Science...

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